Fortunately,
the apparent paradox
found quickly its
origin in an
elementary terminological misunderstanding
(see, e.g., reviews [1]}, [2]}, [3]}
or section  below).
It has been revealed that
the
Bender's and Boettcher's
Hamiltonians \(H\)  are
only non-Hermitian in a “theoretically redundant”,
i.e.,
mathematically preferable but
manifestly unphysical auxiliary Hilbert space
\({\cal H}_{math}\) .
In this sense, these operators have to be
Hermitized in a way explained,
by Scholtz et al,
in review [4]}. In their words,
given a suitable non-Hermitian Hamiltonian \(H\) , one can still work in
an innovated version of quantum theory which
“allows for the normal quantum-mechanical interpretation“.
